![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > 0nnq | Structured version Visualization version GIF version |
Description: The empty set is not a positive fraction. (Contributed by NM, 24-Aug-1995.) (Revised by Mario Carneiro, 27-Apr-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
0nnq | ⊢ ¬ ∅ ∈ Q |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0nelxp 5703 | . 2 ⊢ ¬ ∅ ∈ (N × N) | |
2 | df-nq 10889 | . . . 4 ⊢ Q = {𝑦 ∈ (N × N) ∣ ∀𝑥 ∈ (N × N)(𝑦 ~Q 𝑥 → ¬ (2nd ‘𝑥) <N (2nd ‘𝑦))} | |
3 | 2 | ssrab3 4076 | . . 3 ⊢ Q ⊆ (N × N) |
4 | 3 | sseli 3974 | . 2 ⊢ (∅ ∈ Q → ∅ ∈ (N × N)) |
5 | 1, 4 | mto 196 | 1 ⊢ ¬ ∅ ∈ Q |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 2106 ∀wral 3060 ∅c0 4318 class class class wbr 5141 × cxp 5667 ‘cfv 6532 2nd c2nd 7956 Ncnpi 10821 <N clti 10824 ~Q ceq 10828 Qcnq 10829 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2702 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2709 df-cleq 2723 df-clel 2809 df-ne 2940 df-rab 3432 df-v 3475 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4523 df-sn 4623 df-pr 4625 df-op 4629 df-opab 5204 df-xp 5675 df-nq 10889 |
This theorem is referenced by: adderpq 10933 mulerpq 10934 addassnq 10935 mulassnq 10936 distrnq 10938 recmulnq 10941 recclnq 10943 ltanq 10948 ltmnq 10949 ltexnq 10952 nsmallnq 10954 ltbtwnnq 10955 ltrnq 10956 prlem934 11010 ltaddpr 11011 ltexprlem2 11014 ltexprlem3 11015 ltexprlem4 11016 ltexprlem6 11018 ltexprlem7 11019 prlem936 11024 reclem2pr 11025 |
Copyright terms: Public domain | W3C validator |